# Discrete mathematics and algebra MT3170

This course is intended to give an introduction to the areas of mathematics known as discrete mathematics and the study of modern algebra.

A key aim is to provide an insight into the interactions between these areas, in particular to modern applications such as coding and cryptography.

### Prerequisite

If taken as part of a BSc degree, courses which must be passed before this courses may be attempted:

• MT2116 Abstract mathematics.

### Topics covered

This full course develops the mathematical methods of discrete mathematics and algebra and will emphasis their applications.

• Counting: selections, inclusion-exclusion, partitions and permutations, Stirling numbers, generating functions, recurrence relations.
• Graph Theory: basic concepts (graph, adjacency matrix, etc.), walks and cycles, trees and forests, colourings.
• Set Systems: matching, finite geometries, block designs.
• Abstract groups: revision of key concepts such as cyclic groups, subgroups, homomorphisms and Lagrange’s theorem. Conjugation and normal subgroups. Group actions.
• Applications of algebra to discrete mathematics I: permutations, orbits and stabilisers, the orbit-stabiliser theorem; applications to counting problems.
• Rings and polynomials: the Euclidean algorithm for polynomials, integral domains, ideals, factor rings, fields, field extensions.
• Finite fields: construction, the primitive element theorem, and finite linear algebra.
• Applications of algebra to discrete mathematics II: finite Geometry: designs, affine and projective planes.
• Error-correcting codes: linear codes, cyclic codes, perfect codes.

### Learning outcomes

If you complete the course successfully, you should be able to:

• Demonstrate knowledge definitions, concepts and methods in the topics covered and how to apply these
• Find and formulate simple proofs
• Model situations in a mathematical way and derive useful results.

### Assessment

Unseen written exam (3 hrs).